Overlap–save method

In signal processing, overlap–save is the traditional name for an efficient way to evaluate the discrete convolution between a very long signal $x[n]$ and a finite impulse response (FIR) filter $h[n]$:

where Template:Nowrap for m outside the region Template:Nowrap. This article uses common abstract notations, such as $y(t)=x(t)*h(t),$ or $y(t)=\mathscr{H}\{x(t)\},$ in which it is understood that the functions should be thought of in their totality, rather than at specific instants $t$ (see Convolution#Notation).

Fig 1: A sequence of four plots depicts one cycle of the overlap–save convolution algorithm. The 1st plot is a long sequence of data to be processed with a lowpass FIR filter. The 2nd plot is one segment of the data to be processed in piecewise fashion. The 3rd plot is the filtered segment, with the usable portion colored red. The 4th plot shows the filtered segment appended to the output stream.Template:Efn-ua The FIR filter is a boxcar lowpass with M=16 samples, the length of the segments is L=100 samples and the overlap is 15 samples.

The concept is to compute short segments of y[n] of an arbitrary length L, and concatenate the segments together. That requires longer input segments that overlap the next input segment. The overlapped data gets "saved" and used a second time. First we describe that process with just conventional convolution for each output segment. Then we describe how to replace that convolution with a more efficient method.

Consider a segment that begins at n = kL + M, for any integer k, and define:

$x_k[n]\triangleq \{\begin{array}{cc}x[n+kL],& 1\le n\le L+M-1\\ 0,& \text{otherwise}.\end{array}$

$y_k[n]\triangleq x_k[n]*h[n]=\sum _{m=1}^{M}h[m]\cdot x_k[n-m].$

Then, for $kL+M+1\le n\le kL+L+M$, and equivalently $M+1\le n-kL\le L+M$, we can write:

$y[n]=\sum _{m=1}^{M}h[m]\cdot x_k[n-kL-m]\triangleq y_k[n-kL].$

With the substitution $j=n-kL$, the task is reduced to computing $y_k[j]$ for $M+1\le j\le L+M$. These steps are illustrated in the first 3 traces of Figure 1, except that the desired portion of the output (third trace) corresponds to 1  ≤  Template:Mvar  ≤  L.Template:Efn-ua

If we periodically extend x_k[n] with period N  ≥  L + M − 1, according to:

$x_{k,N}[n]\triangleq \sum _{\ell =-\mathrm{\infty }}^{\mathrm{\infty }}{x}_k[n-\ell N],$

the convolutions  $(x_{k,N})*h$  and  $x_k*h$  are equivalent in the region $M+1\le n\le L+M$. It is therefore sufficient to compute the N-point circular (or cyclic) convolution of $x_k[n]$ with $h[n]$  in the region [1, N].  The subregion [M + 1, L + M] is appended to the output stream, and the other values are discarded.  The advantage is that the circular convolution can be computed more efficiently than linear convolution, according to the circular convolution theorem:

where:

• DFT_N and IDFT_N refer to the Discrete Fourier transform and its inverse, evaluated over N discrete points, and
• Template:Math is customarily chosen such that Template:Math is an integer power-of-2, and the transforms are implemented with the FFT algorithm, for efficiency.
• The leading and trailing edge-effects of circular convolution are overlapped and added,Template:Efn-ua and subsequently discarded.Template:Efn-ua

Pseudocode

(Overlap-save algorithm for linear convolution)
h = FIR_impulse_response
M = length(h)
overlap = M − 1
N = 8 × overlap    (see next section for a better choice)
step_size = N − overlap
H = DFT(h, N)
position = 0

while position + N ≤ length(x)
    yt = IDFT(DFT(x(position+(1:N))) × H)
    y(position+(1:step_size)) = yt(M : N)    (discard M−1 y-values)
    position = position + step_size
end

Efficiency considerations

Fig 2: A graph of the values of N (an integer power of 2) that minimize the cost function $\frac{N({\log }_{2}N+1)}{N-M+1}$

When the DFT and IDFT are implemented by the FFT algorithm, the pseudocode above requires about Template:Nowrap complex multiplications for the FFT, product of arrays, and IFFT.Template:Efn-ua Each iteration produces Template:Nowrap output samples, so the number of complex multiplications per output sample is about:

For example, when $M=201$ and $N=1024,$ Template:EquationNote equals $13.67,$ whereas direct evaluation of Template:EquationNote would require up to $201$ complex multiplications per output sample, the worst case being when both $x$ and $h$ are complex-valued. Also note that for any given $M,$ Template:EquationNote has a minimum with respect to $N.$ Figure 2 is a graph of the values of $N$ that minimize Template:EquationNote for a range of filter lengths ($M$).

Instead of Template:EquationNote, we can also consider applying Template:EquationNote to a long sequence of length $N_x$ samples. The total number of complex multiplications would be:

$N_x\cdot ({\log }_2(N_x)+1).$

Comparatively, the number of complex multiplications required by the pseudocode algorithm is:

$N_x\cdot ({\log }_2(N)+1)\cdot \frac{N}{N-M+1}.$

Hence the cost of the overlap–save method scales almost as $O\left(N_x{\log }_{2}N\right)$ while the cost of a single, large circular convolution is almost $O\left(N_x{\log }_2{N}_x\right)$.

Overlap–discard

Overlap–discard and Overlap–scrap are less commonly used labels for the same method described here. However, these labels are actually better (than overlap–save) to distinguish from overlap–add, because both methods "save", but only one discards. "Save" merely refers to the fact that M − 1 input (or output) samples from segment k are needed to process segment k + 1.

Extending overlap–save

The overlap–save algorithm can be extended to include other common operations of a system:Template:Efn-ua

• additional IFFT channels can be processed more cheaply than the first by reusing the forward FFT
• sampling rates can be changed by using different sized forward and inverse FFTs
• frequency translation (mixing) can be accomplished by rearranging frequency bins

See also

• Overlap–add method
• Circular convolution#Example

References

4. Template:Cite book
5. Template:Cite patent, also available at https://patentimages.storage.googleapis.com/4d/39/2a/cec2ae6f33c1e7/US6898235.pdf

External links

• Dr. Deepa Kundur, Overlap Add and Overlap Save, University of Toronto